∈ (0, ∞) — scale parameter In probability theory, the stable count distribution is the conjugate prior of a one-sided stable distribution.

be a standard stable random variable whose distribution is characterized by

The reason why this distribution is called "stable count" can be understood by the relation

In "Series representation", it is shown that the stable count distribution is a special case of the Wright function (See Section 4 of [4]): This leads to the Hankel integral: (based on (1.4.3) of [5]) Another approach to derive the stable count distribution is to use the Laplace transform of the one-sided stable distribution, (Section 2.4 of [1]) Let

Lambda decomposition is the foundation of Lihn's framework of asset returns under the stable law.

It is derived from the lambda decomposition above by a change of variable such that where This transformation is named generalized Gauss transmutation since it generalizes the Gauss-Laplace transmutation, which is equivalent to

, we arrive at the desired formula: This is in the form of a product distribution.

denotes the PDF of a generalized gamma distribution, whose CDF is parametrized as

term is an exponent representing the second degree of freedom in the shape-parameter space.

It begins with the complementary CDF, which comes from Lambda decomposition: By taking derivative on

For stable distribution family, it is essential to understand its asymptotic behaviors.

This in a way solves the thorny issue of diverging moments in the stable distribution.

(See Section 2.4 of [1]) The analytic solution of moments is obtained through the Wright function: where

is The MGF can be expressed by a Fox-Wright function or Fox H-function: As a verification, at

The word "quartic" comes from Lihn's former work on the lambda distribution[6] where

At this setting, many facets of stable count distribution have elegant analytical solutions.

also becomes a delta function, Based on the series representation of the one-sided stable distribution, we have: This series representation has two interpretations: The proof is obtained by the reflection formula of the Gamma function:

The Wright representation leads to analytical solutions for many statistical properties of the stable count distribution and establish another connection to fractional calculus.

This phenomenon justifies the concept of "floor volatility".

A sample of the fit is shown below: One form of mean-reverting SDE for

[8] This SDE is analytically tractable and satisfies the Feller condition, thus

Extremely low VIX reading indicates a very complacent market.

, carries a certain significance - When it occurs, it usually indicates the calm before the storm in the business cycle.

As the modified CIR model above shows, it takes another input parameter

to simulate sequences of stable count random variables.

The mean-reverting stochastic process takes the form of which should produce

This leads to the well-known quartic stable count result: The ordinary Fokker-Planck equation (FPE) is

The time-fractional FPE introduces the additional fractional derivative

, we obtain the kernel for the time-fractional FPE (Eq (16) of [10]) from which the fractional density

, similar to the "lambda decomposition" concept, and scaling of time